Percents and Applications

💯 Percents and Applications

Part 1 of 5 — What a Percent Really Means

Topics in This Part

🔑 Key Concept: A percent is just a fraction with a denominator of $100$. The word percent comes from the Latin per centum — "per hundred." Master this one idea and every percent problem becomes a fraction problem.

Percent = "Per Hundred"

When you see $45\%$, read it as 45 out of 100:

$45\% = \frac{45}{100} = 0.45$

Picture a $10 \times 10$ grid of $100$ squares. Shading $45$ of them shows $45\%$ — almost half the grid.

💡 A percent can be more than $100$. $100\%$ means "the whole thing," so $150\%$ means "one and a half times as much."

Concept Check 🎯

Percents ↔ Decimals

Because a percent is "out of $100$," converting is all about moving the decimal point two places.

Percent → Decimal: drop the $\%$ sign and move the decimal 2 places left (÷ 100).

$45\% \to 0.45 \qquad 7\% \to 0.07 \qquad 120\% \to 1.20$

Decimal → Percent: move the decimal 2 places right and add a $\%$ sign (× 100).

$0.6 \to 60\% \qquad 0.03 \to 3\% \qquad 2.5 \to 250\%$

⚠️ Watch single digits. $7\%$ is $0.07$, not $0.7$. You must fill an empty place with a zero: $07. \to 0.07$.

Convert Percents and Decimals 🧮

Move the decimal two places.

1) $45\%$ as a decimal $= \,?$ 2) $7\%$ as a decimal $= \,?$ 3) $0.6$ as a percent $= \,?\%$ (enter just the number)

Percents ↔ Fractions

Percent → Fraction: put the number over $100$, then simplify.

$75\% = \frac{75}{100} = \frac{3}{4} \qquad 40\% = \frac{40}{100} = \frac{2}{5}$

Fraction → Percent: divide, then move the decimal $2$ places right (or build an equivalent fraction with denominator $100$).

$\frac{1}{8} = 0.125 = 12.5\% \qquad \frac{3}{4} = \frac{75}{100} = 75\%$

Memorize These Benchmarks

🔑 Knowing these by heart makes mental percents fast: $25\%$ of anything is just one quarter of it.

Match the Forms 🔽

Choose the correct equivalent for each.

💯 Percents and Applications

Part 2 of 5 — Finding a Percent of a Number

🔑 The Core Skill: "$20\%$ of $80$" is a multiplication problem. The word "of" means multiply, and a percent is a decimal in disguise:

$20\% \text{ of } 80 = 0.20 \times 80 = 16$

"Of" Means Multiply

To find a percent of a number, change the percent to a decimal and multiply:

$\text{(percent as decimal)} \times \text{(the number)} = \text{the part}$

Worked Example: $15\%$ of $60$

1. Convert: $15\% = 0.15$
2. Multiply: $0.15 \times 60 = 9$

So $15\%$ of $60$ is $\mathbf{9}$.

Worked Example: $8\%$ of $250$

$8\% = 0.08 \quad\Rightarrow\quad 0.08 \times 250 = 20$

Worked Example: $150\%$ of $40$

$150\% = 1.50 \quad\Rightarrow\quad 1.50 \times 40 = 60$

💡 Because $150\% > 100\%$, the answer ($60$) is bigger than the original number ($40$). That's a good sanity check.

Concept Check 🎯

Estimate to Catch Mistakes

Before you trust a calculation, estimate with a benchmark percent:

• $10\%$ of a number is just that number with the decimal moved one place left: $10\%$ of $60$ is $6$.
• So $15\%$ of $60$ should be a bit more than $6$ — and indeed it's $9$. ✓

💡 If your answer is wildly off your estimate, you probably forgot to convert the percent to a decimal. ($15 \times 60 = 900$ is a giant red flag.)

Find the Part 🧮

Convert to a decimal and multiply.

1) $15\%$ of $60 = \,?$ 2) $8\%$ of $250 = \,?$ 3) $150\%$ of $40 = \,?$

The Three Percent Questions

Every percent problem is one of three types. They all come from the same relationship:

$\text{part} = \text{percent} \times \text{whole}$

Worked Example: "What percent of $24$ is $18$?"

$\frac{\text{part}}{\text{whole}} = \frac{18}{24} = 0.75 = 75\%$

Worked Example: "$30$ is $25\%$ of what number?"

$\text{whole} = \frac{\text{part}}{\text{percent}} = \frac{30}{0.25} = 120$

⚠️ Read carefully. "What percent" → divide part by whole. "Of what number" → divide part by the decimal percent. Mixing these up is the #1 percent mistake.

Pick the Move 🔽

For each question, choose the calculation that solves it.

Decoding the Words

The phrasing tells you which operation to use:

🔑 Translate the words first, then compute. Now solve one of each type yourself.

Solve Each Type 🧮

1) What percent of $24$ is $18$? $\,?\%$ (enter just the number) 2) $30$ is $25\%$ of what number? $\,?$ 3) What is $5\%$ of $200$? $\,?$

💯 Percents and Applications

Part 3 of 5 — Percent Increase & Decrease

🔑 The Idea: When a quantity changes, the percent change tells you how big the change is relative to where it started:

$\text{percent change} = \frac{\text{amount of change}}{\text{original amount}} \times 100\%$

Finding the Percent Change

Two steps:

1. Amount of change $=$ new value $-$ original value.
2. Divide by the original, then convert to a percent.

Worked Example: A price rises from $40$ to $50$

$\text{change} = 50 - 40 = 10$ $\frac{10}{40} = 0.25 = 25\% \text{ increase}$

Worked Example: A price drops from $80$ to $60$

$\text{change} = 60 - 80 = -20$ $\frac{-20}{80} = -0.25 = 25\% \text{ decrease}$

⚠️ Always divide by the original value, not the new one. Going $40 \to 50$ is a $25\%$ increase, but $50 \to 40$ is a $20\%$ decrease — different originals, different percents.

Concept Check 🎯

Markups and Discounts

Stores use percent change constantly: a markup raises a price, a discount lowers it.

There are two equally good methods.

Method A — Find the change, then add or subtract

A shirt costs $25 and is marked up $20\%$:

$\text{markup} = 0.20 \times 25 = 5 \quad\Rightarrow\quad 25 + 5 = 30$

so the new price is $30.

A $80 jacket is $30\%$ off:

$\text{discount} = 0.30 \times 80 = 24 \quad\Rightarrow\quad 80 - 24 = 56$

so the sale price is $56.

Method B — Multiply by the final percent (the shortcut)

• A $20\%$ increase means you keep $100\% + 20\% = 120\%$: $\;1.20 \times 25 = 30$, giving $30.
• A $30\%$ discount means you pay $100\% - 30\% = 70\%$: $\;0.70 \times 80 = 56$, giving $56.

💡 Method B is faster. For a $30\%$-off sale, just multiply by $0.70$ — you pay $70\%$ of the original. One step, same answer.

The One-Step Multiplier 🔽

Choose the single number you'd multiply the original price by.

One Multiplier, Final Answer

You just built every shortcut multiplier. Putting Method B to work:

$\underbrace{1.20}_{+20\%} \times 25 = \$30 \qquad \underbrace{0.70}_{-30\%} \times 80 = \$56$

💡 Method B (multiply once) and Method A (find the change, then add/subtract) always give the same answer. Use whichever you find clearer — but Method B is fewer steps. Try a few on your own.

Markups and Discounts 🧮

Enter the final price in dollars (just the number).

1) A $25 shirt is marked up $20\%$. New price $=$ $? 2) An $80 jacket is $30\%$ off. Sale price $=$ $? 3) A $200 bike is $40\%$ off. Sale price $=$ $?

Working Backward (Finding the Original)

If you know the final price and the percent change, divide to recover the original.

Worked Example: A jacket costs $48 after $20\%$ off. What was the original price?

You paid $80\%$ of the original, so:

$0.80 \times (\text{original}) = 48 \quad\Rightarrow\quad \text{original} = \frac{48}{0.80} = \$60$

⚠️ Don't just add $20\%$ back to $48. That gives $57.60, which is wrong, because the $20\%$ was taken off the larger original price ($60$), not the sale price ($48$). Always divide by the multiplier.

💯 Percents and Applications

Part 4 of 5 — Real-World Applications

🔑 Same skill, new names. Sales tax, tips, commission, and simple interest are all "percent of a number" problems wearing different costumes.

Sales Tax and Tips

Both tax and tip are amounts added on top of a base price.

$\text{tax (or tip)} = \text{rate} \times \text{base price}$ $\text{total} = \text{base} + \text{tax (or tip)}$

Worked Example: $8\%$ sales tax on a $45 purchase

$\text{tax} = 0.08 \times 45 = 3.60$ $\text{total} = 45 + 3.60 = 48.60$

giving a tax of $3.60 and a total of $48.60.

Worked Example: An $18\%$ tip on a $50 meal

$\text{tip} = 0.18 \times 50 = 9.00$ $\text{total} = 50 + 9 = 59.00$

giving a tip of $9.00 and a total of $59.00.

💡 Shortcut: since tax and tip are added on, you can multiply the base by $(1 + \text{rate})$ for the total in one step: $1.08 \times 45 = 48.60$, i.e. $48.60.

Concept Check 🎯

Commission

A commission is a percent of sales that a salesperson earns as pay.

$\text{commission} = \text{rate} \times \text{total sales}$

Worked Example: A realtor earns $6\%$ commission on a $250,000 home

$0.06 \times 250{,}000 = 15{,}000$

which is $15,000.

Worked Example: A clerk earns $5\%$ commission on $1,200 in sales

$0.05 \times 1{,}200 = 60$

which is $60.

💡 Commission is the same operation as tax and tip — "percent of a number" — it's just money earned instead of money added to a bill.

Tax, Tip, and Commission 🧮

Enter dollar amounts (just the number).

1) A $7.5\%$ tax on a $120 item. Tax $=$ $? 2) A $15\%$ tip on a $40 meal. Tip $=$ $? 3) A $6\%$ commission on $1,200 in sales. Commission $=$ $?

Simple Interest

Simple interest is money earned (or owed) on a starting amount, computed with one formula:

$I = P \cdot r \cdot t$

Worked Example: $1,000 at $5\%$ for $2$ years

$I = 1000 \times 0.05 \times 2 = 100$

so the interest is $100.

The total amount in the account is principal plus interest:

$P + I = 1000 + 100 = 1{,}100$

which is $1,100.

⚠️ Two common slips: (1) use $r$ as a decimal ($5\% = 0.05$, not $5$), and (2) keep $t$ in years (6 months $= 0.5$ year).

Simple Interest 🔽

Use $I = P \cdot r \cdot t$ for $500 at $4\%$ for $3$ years.

💯 Percents and Applications

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) convert between percents, decimals, and fractions, (2) find a percent of a number and solve all three percent questions, (3) handle percent increase and decrease, and (4) apply percents to tax, tip, commission, and interest. Let's put it all together.

Quick Reference

⚠️ Remember the big three traps: convert percents to decimals before multiplying, divide by the original for percent change, and use $r$ as a decimal with $t$ in years.

Mixed Review 🔽

From Recognizing to Computing

The review above asked you to recognize the right form. Now let's compute full answers. Keep the playbook handy:

• "What is X% of N?" → multiply.
• "P is X% of what?" → divide $P$ by the decimal.
• A tip total → multiply by $1 + \text{rate}$.

🔑 Decimal-ize the percent first, every time. Then the arithmetic is easy.

Mixed Practice 🧮

1) What is $35\%$ of $200$? $\,?$ 2) $90$ is $75\%$ of what number? $\,?$ 3) A $45 meal with a $20\%$ tip costs a total of $?

Application Challenge 🎯

Exit Quiz ✅

Answer all three to finish the lesson.